How deep do you have to go before you can contribute to the research frontier

Both pictures don't really show how it really is. I am not good at drawing, so I'll try to explain how I see it:
Topics are related not by inclusion, as in your images, but in a directed graph. If $T_1$ and $T_2$ are two topics, then an edge $(T_1,T_2)$ means that $T_2$ depends on $T_1$. It might be an extension of the topic, it might be something that uses techniques and results from $T_1$, there are many ways how one topic can depend on another.
During your studies, you learn a lot of topics that are assumed to have the most outgoing paths, i.e. they are important for many other topics. Your analysis chain would be one such example, although the relation is not that clear, depending on what you do there might be dependencies in both directions.

Now where can we find new topics, unsolved problems and frontiers in this graph? The answer is: everywhere. In every last topic, there are open questions, and there is an infinite number of new topics out there. If you want to do research, you should keep this graph in mind. If your topic is important, in the sense that there are already many outgoing paths known, then chances are high that many people already worked on that, so you have a lot of sources that might help you. However, for the problem to be still unsolved, chances are that it will be really hard to solve it. On the other hand, a topic with few (or no known) outgoing paths might allow you to find results easier, but less people might be interested in them. Thus, it is also part of research to generate outgoing edges from ones topic, to show that it is interesting for questions where it wasn't previously considered.

I myself had a hard time understanding that as a student, I had to live it first. If you want to find the "closest" frontier, I would suggest number theory. Problems like e.g. the Goldbach conjecture can be explained in a single talk, including all the needed information on what a prime number is, but are still unsolved.